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Homomorphisms of 3-Chromatic Graphs

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Homomorphisms of 3-Chromatic Graphs Discrete Mathematics 54 (1985) 127-132 127 North-Holland HOMOMORPHISMS OF 3-CHROMATIC GRAPHS Michael O. ALBERTSON Department of Mathematics, Smith College, Northampton, MA 01063, U.S.A. Karen L. COLLINS Department of Mathematics, M.L T., Cambridge, MA 02139, U.S.A. Received 29 March 1984 Revised 17 August 1984 This paper examines the effect of a graph homomorphism upon the chromatic difference sequence of a graph. Our principal result (Theorem 2) provides necessary conditions for the existence of a homomorphism onto a prescribed target. As a consequence we note that iterated cartesian products of the Petersen graph form an infinite family of vertex transitive graphs no one of which is the homomorphic image of any other. We also prove that there is a unique minimal element in the homomorphism order of 3-chromatic graphs with non-monotonic chromatic difference sequences (Theorem 1). We include a brief guide to some recent papers on graph homomorphisms. A homomorphism f from a graph G to a graph H is a mapping from the vertex set of G (denoted by V(G)) to the vertex set of H which preserves edges, i.e., if (u, v) is an edge in G, then (f(u), f(v)) is an edge in H. If in addition f is onto V(H) and each edge in H is the image of some edge in G, then f is said to be onto and H is called a homomorphic image of G. We define the homomorphism order ~> on the set of finite connected graphs as G ~>H if there exists a homomorphism which maps G onto H. Graph colorings provide the most common examples of homomorphisms: an r-coloring of G is just a homomorph- ism to the r-clique. As a further example note that identifying the antipodal vertices of a dodecahedron is a homomorphism onto the Petersen graph. There is no homomorphism from the Petersen graph to the 5-cycle as any such mapping must identify two non adjacent vertices which necessarily produces a triangle. Graph homomorphisms have arisen separately in three main areas. However, most investigators seem to be unaware of work outside their own area. We present several basic references. The earliest appearance of graph homomorph- isms was in the categorical work of the Prague school. A typical result here is that given any monoid M, there is a graph (cubic graph, k-chromatic graph, etc.) whose endomorphism monoid is M [6]. Computer scientists who study foiiiial languages know of graph homomorphisms in the guise of interpretations of grammars [9-13, 16]; see Salomaa's lovely monograph [13]. A typical result from this area is that given any two (homomorphism) minimal graphs each with more than one edge, then there exists a minimal graph which is naturally between the 0012-365X/85/$3.30 (~ 1985, Elsevier Science Publishers B.V. (North-Holland) 128 M.O. Albertson, K.L. Collins two [16]. Here the order is provided by homomorphism into. To decide whether a graph is 3-colorable is, of course, one of the classic NP-complete problems. It is not surprising that this result has been extended to more general targets. Specific- ally, Maurer, Sudborough, and Welzl have shown that given a fixed odd cycle it is NP-complete to decide if there exists a homomorphism from a graph G to the odd cycle [11]. They conjecture that this result holds for any homomorphism minimal target with more than one edge. Vesztergombi has shown that there exists a homomorphism from a graph G onto the 5-cycle if and only if the chromatic number of the strong product of G and the 5-cycle equals 5 [14, 15]. Given a graph G, the chromatic difference sequence of G, denoted by CDS(G) = a(1), a(2),..., is defined for t = 1, 2,... by l ~, a(j) = a(t, G) = a(t) = maximum # of vertices in an induced j=l t-colorable subgraph of G. This is a generalization of the antichain partition sequence for posets explored by Greene and Kleitman [4, 5]. Albertson and Berman have provided necessary and sufficient conditions for a sequence of no more than 4 terms to be the CDS of some graph [1]. As examples the CDS of the 5-cycle is 2, 2, 1 while the CDS of the Petersen graph is 4, 3, 3. There is a unique graph D shown in Fig. 1 whose CDS is 3, 1, 2. I(A) I(B) H(A) I(C) Fig. 1. Theorem 1. If CDS(G)= a(1), a(2), a(3) and a(2)<a(3), then there exists a homomorphism from G onto D. PreoL Let A, B, and C be the color classes in a fixed 3-coloring of G. Fix a maximum independent set I and let H = G- L Denote by I(A) those vertices in I which are colored A by the fixed 3-coloring. Denote by H(A) those vertices in H which are colored A by the fixed 3-coloring. Similarly define I(B), H(B), I(C), and H(C). The homomorphism f from G to D is indicated by labels assigned to the vertices of D in Fig. 1. Since I is independent and A, B, and C are color classes, f(G) is a subgraph of D. It remains to show that f is a homomorphism onto D. If H were 2-colorable, then a largest color class of H together with I would provide a 2-colorable subgraph of G which would be large enough to prove that a(2)~ a(3). Thus H is 3-chromatic and each of H(A), H(B), and H(C) is not empty. Further the edges in D joining these three vertices must be images of edges in G or H would be 2-colorable. Next we show that I(C) is not empty. If Horaomorphisms of 3-chromatic graphs 129 I(C) were empty, since I is a maximum independent set a(1) = iI(A )l + lI(B)l. Since a(1)+ a(2)>~II(A)I+II(B)I+IH(A)I+IH(B)I, a(2) >~ IH(A )I + IH(B)I. since a(1)+ a(2)> lt(m)l+lr_(B)l+lH(C)l, a(2)>~IH(C)I. But IH(A)I + IH(B)[ + IH(C)I = a(2) + a(3) > 2a(2)>~[H(A)I+IH(B)[+IH(C)[. Similarly I(B) and I(A) are not empty. Finally we need to show that every edge in D is the image of some edge in G. We will actually show more. Suppose that I(C) does not contain a vertex which is joined to both some vertex in H(A) and some vertex in H(B). Let I(C, A) denote those vertices in I(C) which are adjacent to no vertex in H(A). Similarly define I(C, B). Clearly I(C) is the union of I(C,A) and I(C,B). Both I(A)U H(A)U I(C,A) and I(B)U H(B)U I(C,B) form independent sets of vertices. Thus a(1) + a(2)>~ IV(G)I-IH(C)I. Consequently a(3) <~ IH(C)[. Since I and H(C) together form a 2-colorable subgraph of (3, a(2)>-IH(C)l. Thus if D is not a homomorphic image of G, then a(2)>t a(3). [] Suppose there exists a homomorphism from G to H. By definition, the inverse image of an independent set in H is independent in G. Thus any coloring of H pulls back to a coloring of G. One can interpret this remark as saying that a necessary condition for the existen~ of a homorphism from G to H is that the chromatic number of H must be at least as large as the chromatic number of G. Our principal result is an extension of this necessary condition to chromatic difference sequences. For convenience we define the normalized chromatic difference sequence of a graph (3, denoted NCDS(G). If CDS(G)= a(1),a(2),..., then NCDS(G)=c(1),c(2),... where cO) is defined to be a(j)/IV(G)l. The purpose of this is to allow comparisons using the dominance relation. Given two sequences b(1), b(2),..., b(r) and d(1), d(2),..., d(r) the sequence of b's is said to dominate the sequence of d's if t t Y. b(i) Y. a(j) i=1 i=1 for t = 1, 2,..., (r- 1) and equality holds for t = r. We recommend Marshall and Olkin's treatise to the interested reader [8]. 130 M.O. Albertson, K.L. Collins Theorem 2 (The no-homomorphism lemma). If there exists a homomorphism from G to H and H is vertex transitive, then NCDS(G) dominates NCDS(H). ProoL We begin by simplifying some of the notation introduced above. Let a(G) denote the maximum number of vertices in a t-colorable induced subgraph of G. To show that NCDS(G) dominates NCDS(H), we need to show that o~(G)IIV(G)I >I a(H)/IV(H)I. Suppose I(1), I(2),..., I(p) are all of the maximum t-eolorable induced sub- graphs of H. Denote by m the number of these subgraphs which contain a fixed vertex x. Since H is vertex transitive m is well defined. We can express a (H)I[V(H)I explicitly: p p pa(H)= ~ I/(i)1-- Y'.
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